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Properties

Label 1000.4.a.f.1.4
Level $1000$
Weight $4$
Character 1000.1
Self dual yes
Analytic conductor $59.002$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1000,4,Mod(1,1000)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1000.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1000, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1000 = 2^{3} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1000.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,-1,0,0,0,48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(59.0019100057\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 3 x^{9} - 115 x^{8} + 152 x^{7} + 4978 x^{6} + 1245 x^{5} - 90069 x^{4} - 138850 x^{3} + \cdots + 873521 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 5^{5} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.49159\) of defining polynomial
Character \(\chi\) \(=\) 1000.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.55555 q^{3} -1.00777 q^{7} -14.3580 q^{9} -69.8055 q^{11} -19.4611 q^{13} -95.1664 q^{17} -33.1602 q^{19} +3.58319 q^{21} +7.50487 q^{23} +147.051 q^{27} +65.2797 q^{29} +254.468 q^{31} +248.197 q^{33} -349.350 q^{37} +69.1951 q^{39} +256.811 q^{41} -192.903 q^{43} -197.486 q^{47} -341.984 q^{49} +338.369 q^{51} -570.637 q^{53} +117.903 q^{57} +707.268 q^{59} +154.616 q^{61} +14.4696 q^{63} -568.238 q^{67} -26.6840 q^{69} +118.041 q^{71} +930.546 q^{73} +70.3481 q^{77} -661.512 q^{79} -135.180 q^{81} +798.262 q^{83} -232.105 q^{87} -111.956 q^{89} +19.6124 q^{91} -904.774 q^{93} -310.166 q^{97} +1002.27 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{3} + 48 q^{7} + 131 q^{9} + 92 q^{11} - 13 q^{13} - 16 q^{17} + 83 q^{19} + 279 q^{21} + 138 q^{23} + 338 q^{27} + 196 q^{29} + 113 q^{31} - 151 q^{33} - 56 q^{37} + 147 q^{39} + 188 q^{41} + 224 q^{43}+ \cdots + 8897 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −3.55555 −0.684267 −0.342133 − 0.939651i \(-0.611149\pi\)
−0.342133 + 0.939651i \(0.611149\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.00777 −0.0544147 −0.0272073 − 0.999630i \(-0.508661\pi\)
−0.0272073 + 0.999630i \(0.508661\pi\)
\(8\) 0 0
\(9\) −14.3580 −0.531779
\(10\) 0 0
\(11\) −69.8055 −1.91338 −0.956688 − 0.291115i \(-0.905974\pi\)
−0.956688 + 0.291115i \(0.905974\pi\)
\(12\) 0 0
\(13\) −19.4611 −0.415196 −0.207598 − 0.978214i \(-0.566565\pi\)
−0.207598 + 0.978214i \(0.566565\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −95.1664 −1.35772 −0.678860 − 0.734268i \(-0.737526\pi\)
−0.678860 + 0.734268i \(0.737526\pi\)
\(18\) 0 0
\(19\) −33.1602 −0.400393 −0.200197 − 0.979756i \(-0.564158\pi\)
−0.200197 + 0.979756i \(0.564158\pi\)
\(20\) 0 0
\(21\) 3.58319 0.0372341
\(22\) 0 0
\(23\) 7.50487 0.0680380 0.0340190 − 0.999421i \(-0.489169\pi\)
0.0340190 + 0.999421i \(0.489169\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 147.051 1.04815
\(28\) 0 0
\(29\) 65.2797 0.418005 0.209002 − 0.977915i \(-0.432978\pi\)
0.209002 + 0.977915i \(0.432978\pi\)
\(30\) 0 0
\(31\) 254.468 1.47431 0.737157 − 0.675721i \(-0.236168\pi\)
0.737157 + 0.675721i \(0.236168\pi\)
\(32\) 0 0
\(33\) 248.197 1.30926
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −349.350 −1.55224 −0.776119 − 0.630587i \(-0.782814\pi\)
−0.776119 + 0.630587i \(0.782814\pi\)
\(38\) 0 0
\(39\) 69.1951 0.284105
\(40\) 0 0
\(41\) 256.811 0.978223 0.489111 − 0.872221i \(-0.337321\pi\)
0.489111 + 0.872221i \(0.337321\pi\)
\(42\) 0 0
\(43\) −192.903 −0.684126 −0.342063 − 0.939677i \(-0.611126\pi\)
−0.342063 + 0.939677i \(0.611126\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −197.486 −0.612900 −0.306450 − 0.951887i \(-0.599141\pi\)
−0.306450 + 0.951887i \(0.599141\pi\)
\(48\) 0 0
\(49\) −341.984 −0.997039
\(50\) 0 0
\(51\) 338.369 0.929043
\(52\) 0 0
\(53\) −570.637 −1.47893 −0.739463 − 0.673197i \(-0.764920\pi\)
−0.739463 + 0.673197i \(0.764920\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 117.903 0.273976
\(58\) 0 0
\(59\) 707.268 1.56065 0.780325 − 0.625374i \(-0.215053\pi\)
0.780325 + 0.625374i \(0.215053\pi\)
\(60\) 0 0
\(61\) 154.616 0.324533 0.162266 − 0.986747i \(-0.448120\pi\)
0.162266 + 0.986747i \(0.448120\pi\)
\(62\) 0 0
\(63\) 14.4696 0.0289366
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −568.238 −1.03614 −0.518070 − 0.855338i \(-0.673349\pi\)
−0.518070 + 0.855338i \(0.673349\pi\)
\(68\) 0 0
\(69\) −26.6840 −0.0465561
\(70\) 0 0
\(71\) 118.041 0.197309 0.0986544 − 0.995122i \(-0.468546\pi\)
0.0986544 + 0.995122i \(0.468546\pi\)
\(72\) 0 0
\(73\) 930.546 1.49195 0.745974 − 0.665975i \(-0.231984\pi\)
0.745974 + 0.665975i \(0.231984\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 70.3481 0.104116
\(78\) 0 0
\(79\) −661.512 −0.942101 −0.471050 − 0.882106i \(-0.656125\pi\)
−0.471050 + 0.882106i \(0.656125\pi\)
\(80\) 0 0
\(81\) −135.180 −0.185432
\(82\) 0 0
\(83\) 798.262 1.05567 0.527835 − 0.849347i \(-0.323004\pi\)
0.527835 + 0.849347i \(0.323004\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −232.105 −0.286027
\(88\) 0 0
\(89\) −111.956 −0.133341 −0.0666703 − 0.997775i \(-0.521238\pi\)
−0.0666703 + 0.997775i \(0.521238\pi\)
\(90\) 0 0
\(91\) 19.6124 0.0225927
\(92\) 0 0
\(93\) −904.774 −1.00882
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −310.166 −0.324666 −0.162333 − 0.986736i \(-0.551902\pi\)
−0.162333 + 0.986736i \(0.551902\pi\)
\(98\) 0 0
\(99\) 1002.27 1.01749
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1000.4.a.f.1.4 ✓ 10
4.3 odd 2 2000.4.a.w.1.7 10
5.2 odd 4 1000.4.c.c.249.14 20
5.3 odd 4 1000.4.c.c.249.7 20
5.4 even 2 1000.4.a.g.1.7 yes 10
20.19 odd 2 2000.4.a.v.1.4 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1000.4.a.f.1.4 ✓ 10 1.1 even 1 trivial
1000.4.a.g.1.7 yes 10 5.4 even 2
1000.4.c.c.249.7 20 5.3 odd 4
1000.4.c.c.249.14 20 5.2 odd 4
2000.4.a.v.1.4 10 20.19 odd 2
2000.4.a.w.1.7 10 4.3 odd 2